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When I search for 'diversity metrics' it seems like most work has focused on ecological diversity, and I most frequently see the Simpson diversity index. This is equivalent to the economics metric, the Herfindahl–Hirschman Index, for calculating market competition. The limit of these metrics for measuring diversity is that they rely on a single variable per unit, in this case the population share of a biological type or the market-share of a firm.

What is an appropriate metric for quantifying the diversity of a group based on multiple characteristics? Imagine, a group of people and we know their heights, weights, income, race, among other characteristics, and we want to quantify how 'diverse' the group is.

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    $\begingroup$ Some methods for cluster analysis allow a mixture of variable types. I have not encountered anything similar for diversity. The problems start with (1) mixing variable types (2) deciding whether variables should be equally or unequally weighted (3) deciding what to do about correlations (more generally, dependence). $\endgroup$
    – Nick Cox
    Commented Oct 21, 2021 at 18:34

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The concerns raised by @NickCox in the comment above are all good ones. For off-the-shelf "trait diversity" measures, I'd take a look at this unpublished paper by Sam Scheiner: https://doi.org/10.1101/530782. He provides a suite of sensible Hill-number based diversity measures that combine data types (e.g. abundance and trait values, which i think matches your setup). In the appendix, you can find a variety of formulae for to match specific versions of the question of how diverse your groups are.

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